GG is a small satellite which aims at testing the Equivalence Principle (EP) to 1 part in 1017 . It would improve by 4 orders of magnitude the best ground based laboratory tests of the EP as well as those based on Lunar Laser Ranging, and by 2 orders of magnitude the EP test to be performed in space by the μSCOPE satellite. The nature of gravity is fundamental to our understanding of the solar system, the Galaxy and the structure and evolution of the Universe. To date, the experimental evidence for gravitational physics is in agreement with the general theory of relativity; however, there are a number of reasons to question the validity of this theory. Despite the success of modern gauge field theories in describing the electromagnetic, weak and strong interactions, it is still not clear how gravity should be described at the quantum level. In theories that attempt to include gravity, new long-range forces can arise in addition to the Newtonian inverse-square law. Although scalar fields naturally appear in such modern theories, their inclusion predicts a non-Einsteinian behavior of gravitating systems. Predictions have been made that an EP violation might occur at very low level (10-17, 10-18) but possibly also just below 10-12. Such predictions and the overall scenario strongly motivate new searches for very small deviations from general relativity and in particular a possible violation of the equivalence principle since it is at the foundation of the theory. Only an experiment in space allows an EP test of a very high accuracy. Such an experiment requires very weak coupling of the test masses (for high sensitivity), and high frequency modulation (for noise reduction). This is indeed the definition of a supercritical rotor, which however requires two degrees of freedom in order to provide self-centering of the test masses. GG incorporates such a conceptually new accelerometer design (instead of the usual 1D single axis one). This naturally leads to a spacecraft passively stabilized by one axis rotation. Cylindrical symmetry and fast rotation around the symmetry axis make many disturbances unimportant (e.g. radiometer, patch effects etc..), allowing the experiment to be carried out at ambient temperature. The 2D design of the accelerometer allows a full scale ground test to assess the difficulty of the task. GG scientific goal requires detection of a 0.6 picometer relative displacement of the test masses at 1.75⋅10-4Hz (the GG orbital frequency) in 1 week integration time. The GGG prototype supported by INFN in Italy (an apparatus of the same size as required in space, with the same degrees of freedom and the same displacement sensor) has demonstrated long duration, continuous operation (up to 2 months) and has reached the capability to detect a relative displacement of 1 nanometer. This result has been obtained in the presence of noise from motor and bearings (while the experiment in space does not need motor and bearings), from low frequency terrain tilts (which are absent in space), and with a read out electronics based on a 16bit ADC. All these limiting factors can, and will, be reduced. A new 24bit ADC electronics has recently demonstrated a noise level compatible with the GG goal to detect 0.6 picometer in 1 week. The GGG experimental results have increased interest in the GG mission among the scientific community worldwide, and it has now gathered significant support at international level. In Italy, in 2006 GG has been included in the "Piano Aerospaziale Nazionale 2006- 2008" of ASI, where the possibility of launch with a qualification flight of VEGA is considered. A 2-yr Phase A-2 Study with industrial participation has been approved by ASI in 2007 and will start this fall. In summary, GG is a Class M mission in its scientific objective, but it is designed to achieve it with a passively stabilized satellite of low mass (250Kg) and reduced complexity in low Earth orbit (520 km altitude), thus maximizing scientific return per unit cost. It requires a single well defined payload at room temperature based on laboratory tested physical principles, thus greatly reducing risk of failure in space. The spacecraft is based on proven technology except for one item: fine tunable electric thrusters for which a positive solution in the near future is highly desirable for ESA.